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Induction The problem of induction: the problem of explicating the concept of correct inductive evidence (justification for our belief that will make it into knowledge). Correct inductive inference: an inference whose premises, although not necessitating the conclusion, do support it or make it probable. Demonstrative inference: one whose premises necessitate its conclusion, thus cannot fail to be true. Non-demonstrative inference: one which simply fails to be demonstrative - the conclusion is not necessitated by the premises, and is not necessarily truth-preserving. Background - Hume Hume's project in which the problem of induction arises is set against the background of the development of the empirical science of human nature - that is, an attempt to describe the operations of the mind and the interaction of ideas and impressions, and to demonstrate how it is that these mental activities and faculties lead, although not without frequent exception, to true belief. Science sets out to establish general laws and theories that apply to all parts of space and time, and extend beyond the limits of direct observation and perception by virtue of inductive reasoning. Hume showed, however, that if a justification of induction is even possible, it is not easily achieved. His critique set out with an innocuous question: 'how do we acquire knowledge of the unobserved?', and whilst the sciences, in fact, are the most powerful method of doing this we have acquired thus far, it is the philosophical aspect of this question, not the empirical, that is important here: do those beliefs that science helps us to form about the unobserved constitute knowledge?
Basic causal inference: if we have often seen G following F, and we see an F, we will expect to see a G. (Simply introducing probability does not solve the problem. We could adapt basic causal inference to this: 'if we have often seen G following F, and we see an F, it is probable we will see a G'. But while this might divide or complicate inductive habits, it does not eliminate them, and the same questions still need answering!) Uniformity principle Some sort of 'uniformity principle' would establish that the future will resemble the past, and we could employ this principle as a premiss to turn inductive inferences into demonstrative inferences. However, by Hume's own reasoning, such a principle cannot be argued for by demonstrative reasoning, since its denial is non-contradictory. Equally, we cannot suggest that such a principle is confirmed in experience, since we would need the uniformity principle to justify our use of induction, and so cannot use induction to show the truth of the uniformity principle. The obvious circularity here:
Our knowledge from experience is based on the principle of cause and effect The principle of cause and effect is grounded in induction Induction relies on the uniformity principle, that the future will resemble the past We come to know the uniformity principle through experience
Inductive and demonstrative inference Inductive inferences stand opposite demonstrative inferences as our two means of moving from premisses to conclusion. Demonstrative inferences retain truth at the expense of extension of content - the conclusion is necessitated by the premises since it is simply a re-formulation of all or part of the premises' content. These are nonampliative inferences. An ampliative inference has a conclusion with content not implicitly or explicitly present in the premises. Hume's challenge forces us to conclude: we cannot justify any kind of ampliative inference. If it is justified deductively, it is non-ampliative. It if is justified non-demonstratively, it is circular ('uniformity principle'). Induction produces the idea of an effect from the impression of a cause by reason or understanding, by 'a certain association and
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